Interacting quantum walk on a graph

We introduce an elementary quantum system consisting of a set of spins on a graph and a particle hopping between its nodes. The quantum state is build sequentially, applying a unitary transformation that couples neighboring spins and, at a node, the local spin with the particle. We observe the relaxation of the system towards a stationary paramagnetic or ferromagnetic state, and demonstrate that it is related to eigenvectors thermalization and random matrix statistics. The relation between these macroscopic properties and interaction generated entanglement is discussed.